Let me give an example to explain my question more, what would the collection of possible points which satisfy the following condition look like? :
the point's distance from $(-2,0)$ and $(2,0)$ is always equal to 4(which is the distance between $(-2,0)$ and $(2,0)$ itself)?
Well I was pretty sure it would be a pair of rays: one starting from (2,0) and extending infinitely towards the positive x-axis and the other starting from (-2,0) and extending infinitely to the negative x-axis. But when I tried plotting this on a graphing tool, I was surprised that it didn't give any locus.
Why does this happen? is it some personal error of mine while plotting the graph or is it something else?
Solving the equations directly, the solution seems to be $P=(x,0)$ for all $x$, i.e. the locus would be the $x$ axis.
The algebra has to be corrected for sign ambiguity when squaring square roots. Therefore, the part of the axis with $-2\lt x\lt 2$ must be excluded.